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BMO-type norms related to the perimeter of sets. (English) Zbl 1352.46026
The main question answered in the paper under review has its roots in [J. Bourgain et al., J. Eur. Math. Soc. (JEMS) 17, No. 9, 2083–2101 (2015; Zbl 1339.46028)]. This former article contains a proof that certain $$\mathbb{Z}$$-valued functions are constant. The proof is done by introducing a new function space. There the space of functions $$f$$ is considered where the following quantity $$[f]$$, defined as follows, vanishes (integrals are principal values):
$\limsup_{\varepsilon\downarrow 0}\varepsilon^{n-1}\sup_{\mathcal{G}_\varepsilon} \sum_{Q'\in \mathcal{G}_{\varepsilon}} {\,-\hskip-1.08em\int}_{Q'}| f(x)- {\,-\hskip-1.08em\int}_{Q'} f|\, dx,$
where $$\mathcal{G}_\varepsilon$$ denotes a collection of disjoint $$\varepsilon$$-cubes $$Q'\subset Q$$ with sides parallel to the coordinate axes and cardinality not exceeding $$\varepsilon^{1-n}$$. In this earlier paper, it is proved that for measurable sets $$A\subset (0,1)^n$$
$\| 1_A-{\,-\hskip-1.08em\int}_{(0,1)^n} 1_A\|_{L^{n/(n-1)}((0,1)^n)}\leq C[1_A].$
Denoting by $$P(A,(0,1)^n)$$ the perimeter of $$A$$ relative to $$(0,1)^n$$, one has the inequality
$\| 1_A-{\,-\hskip-1.08em\int}_{(0,1)^n} 1_A\|_{L^{n/(n-1)}((0,1)^n)}\leq C\min\{1,P(A,(0,1)^n)\}.$
In the paper under review, the authors look at an isotropic variant of $$[f]$$, since, after all, the perimeter is isotropic as well. They first define $I_\varepsilon(f):=\varepsilon^{n-1}\sup_{\mathcal{F}_\varepsilon}\sum_{Q'\in \mathcal{F}_{\varepsilon}} {\,-\hskip-1.08em\int}_{Q'}| f(x)- {\,-\hskip-1.08em\int}_{Q'} f|\, dx,$ where in opposition to $$\mathcal{G}_\varepsilon$$, cubes in $$\mathcal{F}_\varepsilon$$ are allowed to have arbitrary orientation.
The authors answer the question whether there is a relationship between $$I_\varepsilon(1_A)$$ and $$P(A)$$ by proving that for any measurable set $$A\subset \mathbb{R}^n$$ $\lim_{\varepsilon\to 0} I_\varepsilon(1_A)=\frac{1}{2}\min\{1, P(A)\}.$ This equality also gives a sufficient condition for a measurable set $$A$$ to have finite perimeter, namely $$\lim_{\varepsilon\to 0} I_\varepsilon(1_A)< 1/2$$.
The proof of the main result takes several subsections. Establishing the upper bound $$I_\varepsilon(1_A)$$ is only a few lines. The verification of the lower bound is split according to whether or not the set $$A$$ has finite parameter. Finally, the last part of the proof addresses the one-dimensional setting.
In the last section, some variants are studied. Among them, a localized version of the main theorem is considered, and a new nondistributional characterization of the perimeter is given. The paper ends with a discussion of the relation of a variant of $$[f]$$ to the total variation of $$f$$ and with an appendix which consists of the proof of an isoperimetric inequality first considered in [H. Hadwiger, Monatsh. Math. 76, 410–418 (1972; Zbl 0248.52012)] and later studied by other authors as well, yet not in the needed generality for the paper under review.

##### MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42B35 Function spaces arising in harmonic analysis 28A75 Length, area, volume, other geometric measure theory
##### Keywords:
BMO; perimeter; total variation
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##### References:
 [1] Ambrosio, Functions of bounded variation and free discontinuity problems (2000) · Zbl 0957.49001 [2] Bakry, Lévy-Gromov’s isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math. 123 (2) pp 259– (1996) · Zbl 0855.58011 [3] Barthe, Some remarks on isoperimetry of Gaussian type, Ann. Inst. H. Poincaré Probab. Statist. 36 (4) pp 419– (2000) · Zbl 0964.60018 [4] Bobkov, A functional form of the isoperimetric inequality for the Gaussian measure, J. Funct. Anal. 135 (1) pp 39– (1996) · Zbl 0838.60013 [5] Bobkov, An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space, Ann. Probab. 25 (1) pp 206– (1997) · Zbl 0883.60031 [6] Bourgain, Optimal control and partial differential equations pp 439– (2001) [7] Bourgain, A new function space and applications, J. Eur. Math. Soc. (JEMS) 17 (9) pp 2083– (2015) · Zbl 1339.46028 [8] Brezis, How to recognize constant functions. A connection with Sobolev spaces, Uspekhi Mat. Nauk 57 (4) pp 59– (2002) [9] Dálvila, On an open question about functions of bounded variation, Calc. Var. Partial Differential Equations 15 (4) pp 519– (2002) · Zbl 1047.46025 [10] Giorgi, Nuovi teoremi relativi alle misure (r-1)-dimensionali in uno spazio ad r dimensioni, Ricerche Mat. 4 pp 95– (1955) · Zbl 0066.29903 [11] Federer, A note on the Gauss-Green theorem, Proc. Amer. Math. Soc. 9 pp 447– (1958) · Zbl 0087.27302 [12] Federer, Geometric measure theory (1969) · Zbl 0176.00801 [13] Hadwiger, Gitterperiodische Punktmengen und Isoperimetrie, Monatsh. Math. 76 pp 410– (1972) · Zbl 0248.52012 [14] John, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 pp 415– (1961) · Zbl 0102.04302
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